2tan (problem 3.3.2)

Average Error: 36.8 → 0.4

Time: 11.7s

Precision: 64

Math

\[\tan \left(x + \varepsilon\right) - \tan x\]

↓

\[\left(\left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon} + \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{3}}, \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}\right)}\right)\right) + \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot \sqrt[3]{{\left({\left(\cos \varepsilon\right)}^{3}\right)}^{3}}}\right) \cdot {\left(\cos \varepsilon\right)}^{2}} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}\right)\right) + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right)\]

C

double f(double x, double eps) {
        double r144925 = x;
        double r144926 = eps;
        double r144927 = r144925 + r144926;
        double r144928 = tan(r144927);
        double r144929 = tan(r144925);
        double r144930 = r144928 - r144929;
        return r144930;
}

↓

double f(double x, double eps) {
        double r144931 = eps;
        double r144932 = sin(r144931);
        double r144933 = 1.0;
        double r144934 = x;
        double r144935 = sin(r144934);
        double r144936 = 3.0;
        double r144937 = pow(r144935, r144936);
        double r144938 = pow(r144932, r144936);
        double r144939 = r144937 * r144938;
        double r144940 = cos(r144934);
        double r144941 = pow(r144940, r144936);
        double r144942 = cos(r144931);
        double r144943 = pow(r144942, r144936);
        double r144944 = r144941 * r144943;
        double r144945 = r144939 / r144944;
        double r144946 = r144933 - r144945;
        double r144947 = r144946 * r144942;
        double r144948 = r144932 / r144947;
        double r144949 = 2.0;
        double r144950 = pow(r144935, r144949);
        double r144951 = pow(r144940, r144949);
        double r144952 = r144950 / r144951;
        double r144953 = r144946 * r144943;
        double r144954 = r144938 / r144953;
        double r144955 = r144950 * r144932;
        double r144956 = r144946 * r144951;
        double r144957 = r144942 * r144956;
        double r144958 = r144955 / r144957;
        double r144959 = fma(r144952, r144954, r144958);
        double r144960 = r144948 + r144959;
        double r144961 = pow(r144932, r144949);
        double r144962 = pow(r144943, r144936);
        double r144963 = cbrt(r144962);
        double r144964 = r144941 * r144963;
        double r144965 = r144939 / r144964;
        double r144966 = r144933 - r144965;
        double r144967 = pow(r144942, r144949);
        double r144968 = r144966 * r144967;
        double r144969 = r144961 / r144968;
        double r144970 = r144935 / r144940;
        double r144971 = r144937 / r144941;
        double r144972 = r144970 + r144971;
        double r144973 = r144969 * r144972;
        double r144974 = r144960 + r144973;
        double r144975 = r144946 * r144940;
        double r144976 = r144935 / r144975;
        double r144977 = r144976 - r144970;
        double r144978 = r144974 + r144977;
        return r144978;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.8
Target	15.4
Herbie	0.4

\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

1. Initial program 36.8

   \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm

3. Applied tan-sum 21.3

   \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
4. Using strategy rm

5. Applied flip3-- 21.4

   \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}} - \tan x\]

6. Applied associate-/r/ 21.4

   \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)} - \tan x\]

7. Applied fma-neg 21.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, 1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right), -\tan x\right)}\]

8. Taylor expanded around inf 21.5

   \[\leadsto \color{blue}{\left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)} + \left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon} + \left(\frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{2} \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{3}\right)} + \left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left({\left(\cos x\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)\right)} + \left(\frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{3} \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{2}\right)} + \frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{\cos x \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{2}\right)}\right)\right)\right)\right)\right) - \frac{\sin x}{\cos x}}\]

9. Simplified 0.4

   \[\leadsto \color{blue}{\left(\left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon} + \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{3}}, \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}\right)}\right)\right) + \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{2}} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}\right)\right) + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right)}\]
10. Using strategy rm

11. Applied add-cbrt-cube 0.4

    \[\leadsto \left(\left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon} + \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{3}}, \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}\right)}\right)\right) + \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot \color{blue}{\sqrt[3]{\left({\left(\cos \varepsilon\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}\right) \cdot {\left(\cos \varepsilon\right)}^{3}}}}\right) \cdot {\left(\cos \varepsilon\right)}^{2}} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}\right)\right) + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right)\]

12. Simplified 0.4

    \[\leadsto \left(\left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon} + \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{3}}, \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}\right)}\right)\right) + \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot \sqrt[3]{\color{blue}{{\left({\left(\cos \varepsilon\right)}^{3}\right)}^{3}}}}\right) \cdot {\left(\cos \varepsilon\right)}^{2}} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}\right)\right) + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right)\]

13. Final simplification 0.4

    \[\leadsto \left(\left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon} + \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{3}}, \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}\right)}\right)\right) + \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot \sqrt[3]{{\left({\left(\cos \varepsilon\right)}^{3}\right)}^{3}}}\right) \cdot {\left(\cos \varepsilon\right)}^{2}} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}\right)\right) + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right)\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))